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Q.
The number of ways in which an examiner can assign $30$ marks to $8$ questions, giving not less than $2$ marks to any question is
Permutations and Combinations
Solution:
Let $x_{i}$ denote the marks assigned to the question.
Then $x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}+x_{7}+x_{8}=$
$30$ where $x_{i} \ge2$, $i=1, 2, 3, 4, .......8$
Let $y_{i}=x_{i}-2, i=1, 2, ,...... 8$
$\therefore y_{1}+y_{2}+ ...... +y_{8}$
$=\left(x_{1}+x_{2}+...... + x_{g}\right)-8\times2=30-16=14$
Hence the total number of solutions of this equation is $ ^{14+8-1}C_{8-1}= \,{}^{21}C_{7}$